Option Pricing for Inventory Management and Control
Bryant Angelos, McKay Heasley, and Jeffrey Humpherys
Abstract— We explore the use of option contracts as a means
of managing and controlling inventories in a retail market.
Specifically, merchants can buy option contracts on unsold
inventories of retail goods in an effort to hedge, pool, or
transfer risk. We propose a new kind of European put option
on an inventory where the holder is allowed to freely adjust
the original sale price of the underlying good throughout the
contract period as a means of controlling demand. Assuming
the retailer will chose the profit maximizing pricing policy, we
can price the option accordingly.
I. INTRODUCTION
The capital markets have been recently scrutinized in
the press, congressional hearings, talk shows, etc., due in
part to the active trading of highly complex and yet poorly
understood over-the-couter derivatives and other investment
vehicles used to transfer cash flows and risks in the consumer
debt and mortgage industries. The recent financial crisis
and global recession caused by large-scale holdings of toxic
paper involving collateralized debt obligations, mortgagebacked securities, and subprime mortgages, demonstrates
that we still have much to learn about financial engineering
and the proper quantification of risk and reward. Nonetheless, despite the ever-growing list of blunders and debacles,
businesses in the aggregate enjoy increasingly greater access
to investment capital, less exposure to market risk, increased
liquidity, and higher productivity as a result of derivative
securities and investment banking [5], [7].
In the retail markets, merchants hold inventories of goods
and services in a manner not terribly different than investors
holding portfolios of investment securities. Indeed an investment portfolio can be viewed as an inventory and vice
versa. Our goal is to identify the commonalities between
these two areas and help tie these two industries together, at
least mathematically, under the common umbrella of systems
theory.
There are some obvious differences between the capital and retail markets. One notable example is that the
former is highly instrumented and transactions are usually
cleared centrally by financial intermediaries on a trading
floor, whereas the latter has wildly varying trading practices
and corresponding risks, thus resulting in relatively high
transaction costs, brand effects, and geographically localized
points of sale. Nonetheless as the information age continues
to evolve and consumer purchases become more electronic,
we may expect to see a similar commoditization in the retail
markets over time. There are already web services such
Bryant Angelos and McKay Heasley are research assistants under the
advisement of Jeffrey Humpherys, an assistant professor in the Department
of Mathematics at Brigham Young University, Provo, UT, USA 84602.
Correspondence should be sent to jeffh at math.byu.edu.
as Paypal/eBay, PriceGrabber, and Amazon that centralize
and/or clear numerous retail storefronts thus removing the
need for a direct relationship between the consumer and the
retailer. Similar transaction clearinghouses likewise exist in
the wholesale and shipping industries.
The idea of an option in retail can be mathematically
motivated by the classical newsvendor problem, which we
review in Section II. In this problem, the retailer seeks
to determine the optimal inventory needed to maximize
expected profit when the demand is uncertain, the sale price
is fixed, and the inventory is perishable so that any remaining
items at the end of the day (or whatever the timeframe) are
sold for scrap at some known salvage value; see for example
[8]. Typical examples of perishable goods include bakery
items in a grocery store and magazines on a newsstand,
but electric power, theater tickets, and airplane seats also
fit this description. In fact one could extend this to virtually
any good by using perishability as a proxy for depreciation,
where the future depreciated value of the item is its “salvage
value” under the newsvendor paradigm. We remark that the
solution of the newsvendor problem has many similarities to
the Black-Scholes pricing of an option in the stock market.
Recall that a put option contract gives the owner the right,
but not the obligation, to sell a particular underlying asset at
a given price and at a specified instance (or period) of time.
Hence, the value of a put option depends on the (expected)
future value of the underlying asset. If the asset price rises
beyond a certain level, called the strike price, the put option
becomes worthless. If it goes below the strike price, the put
option can be exercised and the writer pays the buyer the
difference between the strike price and the asset price. In
the case of a retail option, the merchant buys the option
from the writer who then, at the end of the period, pays the
difference between the strike price and the salvage value on
any remaining inventory, or alternatively the contract could
allow or require the writer to pay the strike price and take
physical possession of the unsold inventory; see [2], [3].
One can price such options using the risk-neutral valuation,
where the premium paid by the buyer at the beginning of the
contract period is the future value of the writer’s expected
payout.
In an effort to make retail options more useful, we relax
the constraint on a fixed sale price u, thus allowing the
retailer to freely adjust the sale price of the underlying good
throughout the contract period in order to control demand,
and thus maximize profit. In Section III, we formulate this by
considering demand as a non-homogenous Poisson process
where the expected arrival rate depends on the sale price
of the underlying good. We model our inventory state as
a continuous Markov process, which can be described as a
system of ordinary differential equations (ODEs) that depend
in part on the retailers pricing policy, which if known gives
the risk-neutral valuation of the option contract. In Section
IV, we determine the optimal pricing policy by solving a
dynamic program; see for example [1]. We consider specifically linear and log-linear demand rates, although our method
is quite general. We likewise reduce our dynamic program
to a system of ODEs, which can be solved to provide the
optimal pricing policy. We assume that the retailer will use
the optimal pricing policy, and hence we set the option price
accordingly. Finally, we perform some simulations in Section
V and discuss future directions and offer additional ideas for
using portfolio optimization tools in retail in Section VI.
A. Optimal Inventory Policy
In the classical newsvendor problem, we seek to determine
the optimal initial inventory I0∗ that maximizes profit, given
a retail price u and marginal (or wholesale) cost C0 .
For convenience, we assume X is a continuous random
variable. Let F (x) be the (usually strictly monotone) cumulative distribution function (cdf) and f (x) := F 0 (x) be the
probability density function (pdf) for the demand X. To find
local extrema of E[Π], we take its derivative with respect to
I0 and set it to zero. Hence
d
d
E[Π] = u − C0 − (u − CT )
E[IT ] = 0,
dI0
dI0
or equivalently,
II. MOTIVATING EXAMPLE
In this section we review the classic newsvendor problem.
We consider the task of determining the inventory policy
(or production policy) that will maximize expected profit,
given that one has a good forecast of future demand. More
precisely, we want to know how much inventory I0 our retailer should buy from the wholesaler at cost C0 to maximize
expected profit, knowing that at the end of the sales period
[0, T ] the remaining inventory IT will be salvaged at a price
of CT per unit. We assume a fixed sale price u and that the
demand X is a random variable with known statistics. Since
the demand may exceed the available inventory, we relate
the quantity sold as the random variable
Q = min(X, I0 ).
Hence, the profit Π is the revenue minus the cost of inventory
plus the salvage, or rather
Π = Qu − I0 C0 + IT CT .
(1)
The profit Π is then expressible as the return one would
get from selling out, minus the lost revenue from excess
inventory that is only partially recovered from salvage. This
yields
Π = I0 (u − C0 ) − (u − CT )IT .
(2)
Hence, the expected profit is
E[Π] = I0 (u − C0 ) − (u − CT )E[IT ].
(4)
It is easy to show that the initial inventory I0 that satisfies
(4) is profit maximizing. To compute E[IT ], we define the
indicator random variable
(
1 X ≤ I0
J=
(5)
0 X > I0 .
Then IT = J(I0 − X) and
E[IT ] = I0 E[J] − E[JX].
(6)
Hence, the expected remaining inventory takes the general
form
Z
Z
E[IT ] = I0
f (x) dx −
xf (x) dx.
(7)
x≤I0
x≤I0
Taking the derivative yields
where IT is the remaining inventory, that is, the amount of
product that does not sell and is thus salvaged at a price of
CT at the end of the sale period. This is given by
IT = I0 − Q = max(I0 − X, 0) := (I0 − X)+ .
d
u − C0
E[IT ] =
.
dI0
u − CT
(3)
We remark that in (2), the only random variable in computing
the profit is IT . Hence it suffices in much of our analysis to
compute E[IT ].
d
d
d
E[IT ] = E[J] + I0
E[J] −
E[JX].
dI0
dI0
dI0
It is relatively straightforward to show that E[J] = F (I0 ),
d
E[J] = f (I0 ) and
dI0
d
E[JX] = I0 f (I0 ).
dI0
Hence
d
E[IT ] = F (I0 ).
dI0
Combining with (4), the optimal inventory policy is
u − C0
I0∗ = F −1
.
u − CT
(8)
This is the solution to the classic newsvendor Problem. We
remark that mathematically, the above analysis is very similar
to the derivation of the Black-Scholes pricing of a European
call option; see for example [10].
Option
Value
III. A NEW KIND OF EUROPEAN PUT OPTION
K = strike price
p = option price
K
−p
We present a new type of European put option for retail,
which allows the option holder to adjust prices throughout
the contract period [0, T ] in an attempt to maximize profits.
We present a method for computing the risk-neutral valuation of this retail option, where the initial inventory is m
units. We assume that the demand is a nonhomogeneous
Poisson process, where customer purchases arrive at the rate
λ(u(t)), which depends on the time-dependent sale price
u(t) set by the merchant. Let Ij (t) be the probability that
the inventory level is j units at time t. Then the vector
I(t) = (I0 (t), . . . , Im (t)) is a continuous-time Markov
process, where the transition probabilities likewise depend
on λ(u(t)).
Stock
Price
(a)
Option
Value
I0 = initial inventory
p = option price
I0
−p
Demand
(b)
Fig. 1.
option.
Option value at expiration of (a) a stock option and (b) a retail
Ii,j+1
λ(ui,j+1 )∆t
B. Fixed-Price Retail Option
We now explore the newsvendor Problem from the point
of view of an option contract; see also [2], [3]. Recall that
a European put option gives the holder the right, but not the
obligation to sell an underlying good at a specified price and
time.
Consider a European put option on the remaining inventory IT with strike price K ≥ CT , initial inventory I0 ,
and fixed sale price u. In other words, we have a derivative
contract where the retailer pays p dollars to the option writer
at the beginning of the contract period and then receives
payment of KIT in return for the remaining inventory. If
the retailer sells out, the retailer pays nothing, and the net
option value is −p corresponding to the initial payment at the
beginning of the contract period; see Figure 1. According to
the risk-neutral valuation of the contract, the option is valued
at
p = (K − CT )E[IT ]
(9)
dollars (modulo interest). Moreover the profit for the retailer
is given by
Π = I0 (u − C0 ) − (u − K)IT − KE[IT ].
(10)
We remark that the expected profit in (10) is the same as
that of (2). Indeed options that are based on the risk-neutral
valuation do not improve the expected outcome, but they
do reduce the uncertainty of the outcome. Specifically, the
variance of (10) is given by (u − K)2 V ar(IT ), whereas the
variance of (2) is larger and given by (u − CT )2 V ar(IT ).
The problem with fixed-price retail options is that retailers
may want to change prices throughout the contract period.
If the underlying product is selling faster than expected, the
retailer may be able to increase profits by raising prices.
If the retailer is over stocked, he or she may be better off
lowering prices to stimulate demand. In the next section, we
introduce a new type of European put option that will allow
for dynamic pricing.
Ii,j
Fig. 2.
Ii+1,j
1 − λ(ui,j )∆t
Single period branch describing (11).
A. Discrete-Time Formulation
We discretize the time interval [0, T ]. Let ui,j denote the
sales price when the inventory is at j units at time i∆t,
where i = 0, . . . , n, j = 0, 1, . . . , m, and ∆t = T /n. For
small intervals of time [t, t + ∆t), we assume that an item
sells with probability λ(ui,j )∆t. Let Ii,j = Ij (i∆t) denote
the probability
Pm that the inventory level is j units at time i∆t.
Clearly j=0 Ii,j = 1, Ii,j ≥ 0 for all i, j. Hence, following
Figure 2, the transition probabilities satisfy
Ii+1,j = λ(ui,j+1 )∆tIi,j+1 + (1 − λ(ui,j )∆t)Ii,j .
(11)
Hence, given a pricing policy {ui,j }, we can determine
the expected remaining inventory by computing the finitedifference equations forward to time t = T . Then the
expected remaining inventory is given by
E[IT ] =
m
X
jIj (T ).
j=1
Then the option valuation comes from (9), yielding
p = (K − CT )
m
X
jIj (T ).
(12)
j=1
This is the risk-neutral valuation of the option.
B. Continuous-Time Formulation
We can now take the limit at ∆t goes to zero, thus turning
this problem into a system of ODEs. Simplifying (11), yields
Ii+1,j − Ii,j
= λ(ui,j+1 )Ii,j+1 − λ(ui,j )Ii,j ,
∆t
which, as ∆t → 0, gives the ODE
d
Ij (t) = λ(uj+1 (t))Ij+1 (t) − λ(uj (t))Ij (t),
(13)
dt
where Ij (0) is the initial probability distribution.
Hence, given the pricing policy {uj (t)}m
j=0 , we can compute the state vector I(t) at time T , by integrating the system
of ODEs in (13) on the interval [0, T ]. Figure 3 shows
a sample solution to this differential equation for initial
inventory I10 = 1, and I9 = I8 = · · · = I0 = 0.
0.8
Probability
Ei,j = (1 − λi,j ∆t)Ei+1,j + λi,j ∆t(Ei+1,j−1 + ui,j ), (14)
where λi,j = λ(ui,j ). By taking a derivative with respect to
u, we find that the expected remaining revenue is maximized
when u∗i,j satisfies
1
0.9
0.7
inventory. On each subinterval, the retailer chooses the price
u∗ that will maximize expected remaining revenue on that
subinterval, given the inventory level j. We proceed by
the principle of optimality. Let Ei,j denote the expected
remaining revenue earned during the ith subinterval of time
when the inventory level is j units. If an arrival occurs, the
expected remaining revenue is Ei+1,j−1 + ui,j , and if no
arrival occurs, the expected remaining revenue is Ei+1,j ; see
Figure 4. Thus at each point, we have
0 = λ0 (u∗i,j )(Ei+1,j−1 − Ei+1,j + u∗i,j ) + λ(u∗i,j ).
I=0
I=10
Hence, by solving (15) for u∗i,j , we can compute the optimal
prices and the expected remaining revenues at time i∆t by
using the optimal prices and expected remaining revenues
at time (i + 1)∆t; see Figure 5. This gives us a complete
description of the optimal pricing policy and expected remaining revenues for all subintervals of time and for all
inventory levels.
0.6
0.5
I=7
I=3
0.4
0.3
0.2
0.1
0
0
1
2
3
(15)
4
5
Time
Ei,j
Inventory probabilities over time.
1 − λ(ui,j )∆t
Ei+1,j
2K
j=1
K
j=0
∆t
λ(ui,j )∆t
Ei+1,j−1 + ui,j
Fig. 4.
j=2
{
Fig. 3.
Single period branch describing (14).
IV. OPTIMAL PRICING
In the previous section, we showed how to price an option
for a given pricing policy. We now show how to compute
the optimal pricing policy given a known demand rate λ(u).
We first compute the optimal price by solving a dynamic
program, and then, as in the section above, demonstrate that
the problem can be computed as a solution to a system of
ODEs by taking the subinterval length to zero. We assume
that the retailer will choose the profit maximizing pricing
policy, so that we can give the (Nash optimal) risk-neutral
valuation of the option.
A. Pricing as a Dynamic Program
By partitioning the contract period [0, T ] into n equal
subintervals, as above, we can design a dynamic program
to determine the optimal pricing policy for the underlying
T
0
Fig. 5. Multiple periods for computing the expected remaining revenue.
The rightmost column corresponds to the final period, where the option pays
out K times the remaining inventory. By iterating backwards in time, we
can determine the rest of the grid values for both the optimal price u∗ and
the corresponding expected remaining revenue values.
B. Linear and Loglinear Demand Rates
To more directly connect the expected remaining revenues
and optimal prices, we compute optimal pricing policies for
linear and log-linear demand rates.
1) Linear Demand: If the parameter of demand λ(u) is
linear with respect to price, that is λ(u) = b − au for some
a, b > 0, then (15) gives
Ei+1,j − Ei+1,j−1
b
+ .
2
2a
2) Loglinear Demand: We repeat the process described
above but with a different demand rate. We let λ(u) = au−b ,
b > 1. Then (15) gives
u∗i,j =
u∗i,j =
b
(Ei+1,j − Ei+1,j−1 ).
b−1
becomes very large, one could treat inventory levels as a
continuous variable, and resort to stochastic PDEs, as is done
with option pricing in the stock market (despite the fact that
the underlying also takes on discrete prices).
As a check, we generated simulations to realize consumer
purchases and examine how the price of both the underlying
and the option move over time; see Figure 6. We randomly
generated numbers on the unit interval at each time step, and
then dropped the inventory by one if the number generated
was less than λ(ui,j ). We then computed the option price at
that moment. Notice that both images seem to behave like a
random walk, and look similar to stock-market data.
Finally, we describe how the option price varies as we
toggle parameters. In Figure 7 we plot the option price
against the initial inventory and the strike price, respectively.
These vary in a monotonic, nonlinear fashion, as expected,
growing linearly in the infinite limit and converging to zero
for small values. We remark that the option price for varying
contract periods has an inverse relationship to how the option
price varies with initial inventory, that is, small contract
periods price as if the initial inventory were large, and large
contract periods price as if the initial inventory were small.
2.62
2.6
Sale Price
2.58
2.56
2.54
2.52
2.5
0
200
400
600
800
1000
Time
(a)
16
14
Option Price
12
10
80
8
70
6
4
0
200
400
600
800
Option Price
60
1000
Time
(b)
Fig. 6.
Realization of (a) the optimal price of the underlying asset
throughout contract period and (b) the option price throughout the contract
period.
50
40
30
20
10
0
100
C. Pricing in Continuous Time
200
250
300
Initial Inventory
We conclude this section by showing that the optimal
prices and expected remaining revenues can be computed
by solving a certain system of ODEs much as we did in the
previous section. By simplifying (14), we get
(16)
(a)
450
400
350
Option Price
Ei+1,j − Ei,j
= λ(ui,j )(Ei+1,j − Ei+1,j−1 − ui,j ),
∆t
which, by letting ∆t → 0, yields the ODE
d
Ej (t) = λ(uj (t))(Ej (t) − Ej−1 (t) − uj (t)),
dt
where E0 (t) = 0 and Ej (T ) = jK.
150
300
250
200
150
100
V. SIMULATIONS
We can code up our two systems of ODEs in MATLAB
in just a few lines of code. For the integration, we use
ode45. This variable-step routine allows for large steps
when the errors are small, thus significantly speeding up
computation time. Most of our runs executed in under a
second. For large inventories or contract periods, the runtime
increased proportionately. In the case that the inventory
50
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Strike Price
(b)
Fig. 7.
price.
Option price as we vary (a) the initial inventory and (b) strike
VI. INVENTORY VS. PORTFOLIO MANAGEMENT
From an inventory manager’s point of view, it can be
difficult, when looking at raw transactions data, to know
whether low sales are due to low demand (e.g., nobody is
buying) or low supply (e.g., retailer is out of stock). A retail
option allows one to easily differentiate between low sales
and low supply. Indeed the price of the option goes to zero
when the inventory runs out whereas low sales would result
in the option value being high. By using options and other
financial instruments in a retail environment, merchants can
better understand their markets by treating their inventories
as money managers do portfolios. However in retail, there
is the added advantage of being able to dynamically control
the price of the underlying goods in an attempt to maximize
profits.
There are many directions to take this portfolio view of
inventory theory and control. Several recent results exist for
managing portfolios of stock options over multiple periods;
see for example [9]. We could also explore risk-management
strategies ranging from a myriad of techniques including
value-at-risk, cash-flow-at-risk, etc. Another direction is the
further use of more exotic options. Variations on floors,
caps, Asian options, Bermuda options, etc., should all be
considered to see if there are “retail possibilities”.
Another area of interest is how to price an option when
dealing with an incompetent retailer. Note that the price of
the option proposed here requires that the retailer continually
adjust prices to maximize profits. However, if the retailer
isn’t adjusting them optimally, then the writer will be forced
to buy more unsold inventory than necessary at the end of the
contract period. While the existence of a Nash equilibrium
should preclude such events theoretically assuming symmetric information, market efficiency, etc., in practice prices may
be set badly. Room for such incompetence may need to be
priced in the option.
We remark that options have been around for centuries
in the financial markets [11] (most notably over the past
30 years), and they have only recently been suggested for
use in a retail environment [2], [3]. There is, however, an
extensive supply chain contracts literature going back for
decades; see [12] for a review, as well as a real options
literature in business strategy; see [6] and references within.
It would be worthwhile to pull all of this literature together
under the common umbrella of systems theory. Indeed recent
events demand better analysis of financial modeling and a
deeper understanding of the dynamics of the markets.
Finally, it is unclear whether retail option contracts could
ever materialize into exchange-traded instruments like options on stocks, futures, etc. Surely some could find a natural
home in certain niche retail markets, but overall the lack of
standards, trust, third-party or government regulation, and/or
difficulties in instrumentation may make it difficult to reliably
create an active market for retail options in a general setting.
We remark, however, that the use of retail options as a
means of centralized inventory management and control in
a retail chain, or more generally an organized network of
manufactures, wholesalers, distributors, retailers, speculators,
and financiers, may prove to be valuable. Indeed, retailers
can essentially operate as portfolio managers. Retail chains
can buy and sell options internally, between their distribution
centers, individual stores, and their own corporate finance
department can act as the hedge fund for the company by
pooling risks much like an insurance company. Moreover, by
using the put option proposed in this paper, where the prices
can be adjusted dynamically, the retailers can then use the
price to control the returns in their portfolio, both for an
individual store and in the aggregate.
VII. FUTURE WORK
We are currently considering variations on this work. To
make our retail option viable, we need to further explore
the degree to which the option price is sensitive to input
parameters. We also need to flesh out the details about the
variance in the option payoff.
We are also interested in seeing if a set of options can
be written on an underlying collection of inventory, whether
is is reasonable to have variable payoffs depending on how
much inventory is remaining, and how the option might work
with multiple periods and multiple echelons. Other issues
include state estimation of the arrival rate, the effects of
substitutes and complements, and the effects of coupons and
other manufacturer-based promotions on the option price.
VIII. ACKNOWLEDGMENTS
This work was supported in part by the National Science Foundation, Grant No. DMS-0639328, BYU mentoring
funds, and a gift from Park City Group, Inc. The authors
thank PCG President and CEO Randall Fields and his associates for useful and stimulating conversations throughout
this work.
R EFERENCES
[1] D. Bertsekas. Dynamic Programming and Optimal Control, (Volumes
1 and 2). Athena Scientific, 2005.
[2] A. Burnetas and P. Ritchken. Option pricing with downward-sloping
demand curves: The case of supply chain options. Management
Science, 51(4):566–580, 2005.
[3] F. Chen and M. Parlar. Value of a put option to the risk-averse
newsvendor. IIE Transactions, 39(5):481–500, 2007.
[4] J. Cox, S. Ross, and M. Rubinstein. Option Pricing: A Simplified
Approach’. International Library of Critical Writings in Economics,
143:461–495, 2002.
[5] A. Demirguc-Kunt and R. Levine. Stock markets, corporate finance,
and economic growth: An overview. World Bank Economic Review,
10(2):223–39, May 1996.
[6] A. Dixit and R. Pindyck. Investment under uncertainty. Princeton
University Press Princeton, NJ, 1994.
[7] R. Levine and S. Zervos. Stock markets, banks, and economic growth.
American Economic Review, 88(3):537–58, June 1998.
[8] E. L. Porteus. Foundations of Stochastic Inventory Theory. Stanford
University Press, Stanford, 2002.
[9] J. Primbs. Dynamic hedging of basket options under proportional
transaction costs using receding horizon control. Preprint, 2007.
[10] S. M. Ross. An introduction to mathematical finance. Cambridge
University Press, Cambridge, 1999.
[11] B. Smith. A History of the Global Stock Market: From Ancient Rome
to Silicon Valley. University Of Chicago Press, 2004.
[12] A. A. Tsay, S. Nahmias, and N. Agrawal. Modeling supply chain
contracts: a review. In S. R. Tayur, R. Ganeshan, and M. J. Magazine,
editors, Quantitative Models for Supply Chain Management. Kluwer
Academic Publishers, Norwell, MA, USA, 1998.